Tying the Knot

There are 6 strings clustered together. One end of each string is at point B (the top), and the other at point A (the bottom). First, two of the ends at point A (randomly) are tied together. Then the two more are tied together, and then the last two. Next, two ends at point B are tied together. Then the two more are tied together, and then the last two.
What is the possibility that all the strings will be tied together in one large loop?

Note: Simplify the answer.

Hint

There are three possible combinations of loop sizes that the strings can form.Hide

Answer

The possibility that the strings will form one large loop is 8 in 15 times, or 8/15.
You determined the probability of each situation by finding out the probability of many events. You only had to deal with the bottom knots, because if you left out the top knots, you will just be simplifying the problem a little bit. Then you figure out all the possibilities for the first knot (of the bottom) that could be tied. Then, using those, you found out the three possibilities for each basic diagram, having only the first knot. That gives you 45 possible knot combinations, which is every combination for the bottom half. Then you simply count out the different groups (1 loop, 2 loops or 3 loops) and simplified it. There are 24/45 for 1 loop, 18/45 for 2 loops, and 3/45 for 3 loops. Simplified that is:
**1 Loop: 8/15**
2 Loops: 6/15
3 Loops: 1/15Hide